As mentioned in the description these drop fairly regularly in biomare, especially from the floaters and bloaters.
If you are unsure how to get to biomare, you can check out our guide here (as well as our guide to getting through biomare: here)

However you don't need specifically to go there, as the quality of the gems is irrelevant, so if the mobs are giving you trouble or it's too camped, you can easily search elsewhere. It's worth noting that the mission won't update until you actually start to hand the gems in, so don't worry that it's still on 1/5.

The backyards are a fairly good source of gems. One of the advantages to getting these lower quality gems is that they will be much easier to cut.

This process takes x5 in Mechanical Engineering and x1.9 in Agility Skills based on the QL of the gem.

note: if you loose or accidentally delete the gem cutter, they are available from the Electrical and Mechanical Engineering booths in shops.

If you don't have the skills to do the cutting yourself, then you can simply hand them to a friendly tradeskiller to cut for you.
Now that you've got your 5 cut gems it's time to head to Mr. Carlsberg who can be found in the agency building.

Mr. Carlsberg doesn't do gem cutting
but if he did, it would probably be the best gem cutter in the world.

Opening a conversation with him and answering "I have the perfectly cut stones for you" will enable you to give him the gems one at a time (this will also update the mission), once you have given him all 5 the mission will complete. You even get to keep the gem cutter.